Concept of random sample? I have a truly problem understanding it. I have to solve a probability problem and it says that we take a random sample of size 10. But I don´t understand the concept (I´m on my first course on probability). 
Suppose that we have a box with 100 balls and I take a random sample of size 10
Is a random sample of size 10 if


*

*I take AT THE SAME TIME 10 balls? or

*I take one ball, then return it and repeat this process ten times?


Thanks for your help.
 A: This is a good question, and highlights the issue with what we call randomness. In a question such as this, the best way to think about it is that a person would randomly take a single ball out, then randomly take another one out, and so on and so forth, until they have a sample of 10 balls. This is called sampling without replacement. However, suppose we had the balls numbered from 1 to 100. In this case, we could take one ball out, note the number and then place it back; take another out, note the number, and place it back. This process is called sampling with replacement. 
However, generally, when a text states that a sample is taken from a population, it would explicitly mention it if there was sampling with replacement and normally, if it is not stated explicitly, it is assumed that the sample was taken without replacement.
A: Aside from what they're called, both of your examples are random sampling if each ball is randomly chosen. There is more to randomness than with or without replacement. For example, we can draw balls until some criterion is reached. Or the balls sampled need not all be distinct. Or if the balls are small, we could reach in and grab a handful so that the number of balls grabbed is also random. We can also randomly sample items that are not equally likely to be sampled. And there are more involved ideas of random sampling. The analysis depends on the kind of sampling as you'll see when you learn more. The more complicated notions of sampling are encountered more in statistics than in probability theory. "Random" simply means that there is an element of chance in the selection, i.e., it is not systematic.
